IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 3, MARCH 2017 383
Noncircular Measurement and Mitigationof I /Q Imbalance for OFDM-Based
WLAN TransmittersZhe Li, Student Member, IEEE, Yili Xia, Member, IEEE, Wenjiang Pei, Kai Wang,
Yongming Huang, Member, IEEE, and Danilo P. Mandic, Fellow, IEEE
Abstract— In future high-speed communication networks, thein-phase/quadrature (I / Q) imbalance mitigation and oscillatordrift compensation is a key issue in the design of orthog-onal frequency division multiplexing (OFDM)-based wirelessLAN (WLAN) transmitters. To this end, we propose a two-stage I / Q imbalance measurement method, where by virtue ofthe WLAN standard-compliant training sequences, a coarse I / Qimbalance estimation is initially performed jointly with channelequalization. This makes it possible to decouple the effects offrequency-selective channels from the exact amplitude and phaseimbalances induced by the local oscillator. Next, the so recoveredsymbols in DATA field of standardized OFDM systems, such asthe IEEE 802.11ac, are recalibrated using a decision-directedscheme; this facilitates least squares-based fine I / Q imbalanceestimation. For rigor, augmented complex statistics is employedto account for the effects of data noncircularity and widely linearnatures of communication channels. Computer simulations andreal world experiments based on the IEEE 802.11ac compliantsignals demonstrate the high accuracy of the proposed techniquefor OFDM-based WLAN transmitters.
Index Terms— in-phase/quadrature (I / Q) imbalance, orthogo-nal frequency division multiplexing (OFDM), RF measurements,transmitter testing, wireless LAN (WLAN).
I. INTRODUCTION
ORTHOGONAL frequency division multiplex-ing (OFDM) techniques are widely adopted in
current wireless LAN (WLAN) standards, such as the IEEE802.11 a/g/n/ac [1]–[4]. An efficient implementation ofsuch physical layers is challenging, owing to the problemsarising from current consumer RF integrated circuits of
Manuscript received May 11, 2016; revised August 10, 2016; acceptedSeptember 26, 2016. Date of publication January 5, 2017; date of cur-rent version February 8, 2017. This work was supported in part by theNational Natural Science Foundation of China under Grant 61271058 andGrant 61401094, in part by the Natural Science Foundation of JiangsuProvince under Grant BK20140645, in part by the Fundamental ResearchFunds for the Central Universities under Grant 2242016K41050, and inpart by the Scientific Research Foundation for the Returned Overseas Chi-nese Scholars, State Education Ministry of China. The Associate Editorcoordinating the review process was Dr. Matteo Pastorino. (Correspondingauthor: Yili Xia.)
D. P. Mandic is with the Department of Electrical and ElectronicEngineering, Imperial College London, London SW7 2AZ, U.K. (e-mail:[email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIM.2016.2639778
the WLAN transmitter (TX). In particular, imperfectionsof local oscillators (LOs) in RF circuits, at the analogfront end, cause the in-phase/quadrature (I /Q) imbalancewhich is characterized by differences in the amplitudesof I /Q oscillators, together with the phase shift from thenominal 90◦ [5]. The I /Q imbalance also introducesthe so-called mirror frequency interference (MFI), whichimpairs the modulation accuracy of the transmitter RFsignal and considerably degrades the overall performanceof the transmission system. Critically, the impact of I /Qimbalance is more pronounced in systems which employhigh-order modulations and high coding rates, thismakes the effective I /Q imbalance measurement andtroubleshooting a fundamental task in the design of highdata-rate communication testing systems.
The LO-induced I /Q imbalances can be assumed constantover the signal bandwidth, while a transmitter may also exhibitfrequency-selective I /Q imbalances, caused by a mismatchin the baseband reconstruction filters [6]. Due to the com-plexity of RF circuit design and fabrication, the effect of theLO-induced I /Q impairments is more significant than that ofthe I /Q imbalances caused by filter mismatch in basebandcircuits [7]. Most current studies dealing with frequency-dependent I /Q impairments in transmitters [6], [8], [9] employa feedback circuit from the RF to the baseband in TX, inorder to perform an online calibration. Due to the complicatedand frequency-dependent I /Q imbalance model, intertwinedwith the channel impulse responses of the transmitter and theauxiliary feedback loop, it is difficult to provide an explicitestimation on the exact degree of I /Q imbalance. However,from the point of view of chip manufacturers, it is desirable toseparate the measurement and calibration stages so as to betterunderstand the I /Q impairments within the transmitters undertest. To that end, in this paper, we focus on the enhancementof measurements of LO-induced frequency-independent I /Qimpairments at the instrumentation level.
Measurement methods for the evaluation of frequency-independent I /Q imbalance are extensively reportedin the literature, and can be classified into envelopedetector (ED)-based and demodulator-based, according to thedetection types of the modulator output signal. In analog EDs,the detection can be aided by a series of auxiliarysinusoidal signals [10]–[12], or by access to the transmittedI /Q signals [13]. By using an ideal quadrature demodulator,
384 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 3, MARCH 2017
a clustering-based method in [14] detects the I /Q impairmentsof the modulator and consists of three stages: signaldemodulation, constellation clustering (to match each I /Qdiagram symbol to its ideal position), and evaluation ofthe amount of impairment. However, this method cannotbe directly applied to multicarrier OFDM transmitters.An analytical model which accounts for the way I /Qimpairments affect the RF output signal was proposed in [15]in the context of generic OFDM transmitters. An approachthat extracts data with asymptotic amplitude normalizationwas discussed in [16] for OFDM WiMAX standard-complianttransmitters. Most quadrature demodulator-based methodsdiscussed earlier, perform channel equalization before I /Qimbalance estimation, which implicitly assumes an idealbaseband measurement channel. However, in a real world I /Qmeasurement setup: 1) the physical channel connecting thetransmitters under test and the measurement testbed,e.g., a vector signal analyzer (VSA) [17], is usually connectedvia a coaxial cable; 2) the perfect reconstruction low-passfilters in I /Q branches of transmitter–receiver chain oftenrequire linear-phase for zero intersymbol interference;these result in a frequency-selective channel impulseresponse [18]. Consequently, at instrumentation level forwideband multicarrier systems, the I /Q imbalance distortionintroduced by the channel varies for different subcarriers.It is therefore desirable to perform joint I /Q imbalancemeasurement and channel equalization, so as to remove thechannel effect from the measured I /Q imbalance withintransmitters.
Recently, the blind I /Q imbalance estimation/compensationmethods based on the second-order signal statistics havebecome particularly attractive, due to their relatively lowimplementation complexity. Based on the proper (second-ordercircular) statistical behavior of the desired signals [19]–[22],such methods include eigenvalue decomposition-assistedwhitening transforms [23], block-based self-image can-cellers [24], and adaptive filtering algorithms based on thewidely linear estimation model [25]. Despite the potentiallyhigh I /Q compensation accuracy, such methods are associatedwith the amplitude and phase ambiguities. This is becausethe main aim of I /Q imbalance compensation is to eliminatethe MFI component from the I /Q imbalanced transceiversignal. However, the MFI-eliminated signal is a complex-valued scaled version of the I /Q imbalance-free (desired)transceiver signal, and this complex-valued scaling factorintroduces both amplitude scaling and phase rotation effectson the constellation mapping. By taking into consideration, thefrequency-selective nature of the baseband equivalent channel,standard second-order signal statistics cannot provide enoughdegrees of freedom to estimate this complex-valued scalingfactor, which in fact contains useful information on both theamplitude and phase imbalances; we therefore need to designenhanced methods for exact I /Q imbalance measurement atthe transmitter side.
In this paper, we propose a two-stage framework forthe I /Q imbalance measurement at the instrumentation levelof OFDM-based WLAN transmitters. In the initial stage,a block-based blind method, which fully exploits the available
second-order signal statistics, is proposed. Then, making useof the WLAN standard-compliant training sequences, coarseI /Q imbalance estimation is performed jointly with channelequalization in order to decouple the effects of baseband mea-surement channel on the exact amplitude and phase imbalancesinduced by the LO. Finally, the initially recovered symbols inDATA field of standardized OFDM systems, such as the IEEE802.11ac, are recalibrated using a standard decision-directedscheme, which facilitates fine I /Q imbalance estimation. Theperformance of the proposed I /Q imbalance measurementmethod is assessed through numerical simulations and realworld experiments at the instrumentation level. This yields per-formance advantages over the least squares (LSs) method [26]for OFDM-based WLAN transmission systems, especiallywhen higher order modulation and coding schemes (MCSs)are employed. The main contributions of this paper are:1) unlike the existing demodulator-based I /Q imbalance mea-surement/troubleshooting methods [14]–[16], [27], the equiv-alent baseband measurement channel is considered to befrequency-selective, which represents a more practical andgeneric measurement scenario; 2) apart from the commonlyused the second-order circularity (properness) condition [24],we make use of the full second-order statistics of the desiredsignal by considering the pseudocross correlation between asubcarrier and its mirror frequency counterpart within OFDMtransmission systems.
The rest of this paper is organized as follows. Section IIgives a brief on the mathematical modeling of I /Q imbal-ances in OFDM-based WLAN transmitters and the fullsecond-order statistics of complex-valued random signals. Theproposed method is described in Section III. Simulationsand real world measurement experiments are given inSections IV-A and IV-B, respectively. Section V concludesthis paper.
II. PRELIMINARIES
A. I /Q Imbalances in OFDM-Based WLAN Transmitters
Within the OFDM-based transmitters, frequency-independent I /Q imbalances induced by the LO are amajor obstacle in practical transmitter calibration. Fig. 1shows a typical I /Q imbalanced transmitter, where therelative amplitude and phase imbalances between the I and Qchannels are denoted by g and θ , and represent mismatchesof the quadrature mixer circuit. The discrete-time basebandequivalent modulator output signal x(n) is then typically ofthe form [24]
x(n) = βs(n) + αs∗(n) + d (1)
where s(n) denotes the discrete-time desired (I /Q imbalance-free) baseband waveform, d is the dc-offset component, andβ and α are, respectively, defined as
β = 1/2(1 + ge− jθ )
α = 1/2(1 − ge− jθ ). (2)
Note that due to the complex-valued nature of β and α, theexact values of both the amplitude and phase imbalances gand θ can be directly calculated through a ratio between βand α, e.g., (β)/(α), instead of using their exact values.
LI et al.: NONCIRCULAR MEASUREMENT AND MITIGATION OF I /Q IMBALANCE FOR OFDM-BASED WLAN TRANSMITTERS 385
Fig. 1. Architecture of an I /Q imbalanced modulator.
B. Second-Order Signal Statistics
Since in OFDM systems, the output signals are transmittedby blocks, it is natural to interpret the imbalance model in (1)in the frequency domain, to yield
Xk(m) = βSk(m) + αS∗−k (m) (k �= 0) (3)
where k and m refer, respectively, to the subcarrier index andbaseband OFDM symbol index, Sk(m) represents the desireddata on subcarrier k within the mth baseband OFDM symbol,and Xk(m) is the resulting observation of Sk(m), affected bythe I /Q imbalance.1
We follow the standard assumptions that the desiredsignal Sk(m) is a zero-mean ergodic proper randomprocess with equal variances, σ 2
s , in each subcarrier k,for which the distribution exhibits circular symmetry(i.e., E[Sk(m)Sk(m)] = 0) [19]. The second-order circular-ity (properness) assumption is valid in most cases withinOFDM-based WLAN standard family based on complex-valued modulations, such as the M-quadrature-amplitude mod-ulation (QAM) used in the IEEE 802.11 a/g/n/ac [1]–[4].Furthermore, in such cases, there exists no mutual depen-dence between the transmitted signals in the subcarrier andits mirror frequency counterpart, i.e., E[Sk(m)Sk
∗(m)] =E[S−k(m)S−k
∗(m)] = σ 2s and E[Sk(m)S−k(m)] = 0 [28].
Then, from (3), the autocorrelation at subcarriers k and −k isgiven by
E[Xk(m)X∗k (m)] = E[X−k(m)X∗−k(m)]
= (|β|2 + |α|2)σ 2s (4)
and the pseudocross correlation by
E[Xk(m)X−k(m)] = 2βασ 2s . (5)
Note that this pseudocross correlation vanishes only when themodulator output is I /Q imbalance-free, i.e., α = 0 in (3),this will play a key role in the proposed I /Q imbalancemeasurement framework.
1Note that, compared with (1), the dc-offset component d has been removedfrom the analysis, since the subcarrier 0 has been selected as one of the null-subcarriers in OFDM-based WLAN standards due to practical implementationissues.
III. PROPOSED SECOND-ORDER STATISTICS-BASED I /QIMBALANCE MEASUREMENT
In wireless indoor or outdoor communication scenarios,the transmission channel is usually considered as a wide-sense stationary uncorrelated scattering random process, andhence, its impulse response is usually assumed to be timevarying and exposed to frequency-selective fading. There-fore, the properness (second-order circularity) assumption onthe received signal is always valid, no matter whether thetransmitted signal Xk(m) is proper or not. In other words,a time varying fading channel actually reinforces the propernature of a signal [25]. However, in real world I /Q imbalancemeasurement setups for OFDM-based WLAN transmitters,the physical channel connecting transmitters under test andthe measurement test bed, e.g., a VSA [17], is usually in theform of a coaxial cable, while other RF or baseband com-ponents in the transmitter–receiver chain are considered rel-atively time-invariant within one OFDM transmission frame.Moreover, the perfect reconstruction low-pass filters in the I /Qbranches of the transmitter–receiver chain often require lin-ear phase for zero intersymbol interference, resulting in afrequency-selective channel for the measurement [18]. As aresult, in wideband multicarrier systems, the time-invariant andfrequency-selective property of the channel impulse responsestill maintains the impropriety of the transmitted I /Q imbal-anced signals [23], but the impropriety-related distortion bythe frequency-selective channel varies for different subcarriers.Therefore, it is desirable to perform joint I /Q imbalancemeasurement and channel equalization so as to decouple thechannel effect from the exact I /Q imbalance amount withintransmitters.
A. Baseband Signal PreprocessingIn order to retrieve the baseband components of the
modulator output signals at a testbed, e.g., a VSA, several sig-nal preprocessing operations need to be performed, accordingto standard requirements. As shown in Fig. 2, the initial oper-ations for the acquisition of the baseband components includepower triggering, automatic gain control (AGC), resampling,and subframe recognition [16]. The data acquisition is trig-gered when the received power exceeds a reference threshold.An AGC algorithm is then used to adjust both the low noiseand variable gain amplifiers in AGC circuits in order tomaintain the same level of amplification for input signals.An ordinary digital resampling algorithm is then applied onboth the I /Q components, extracted by the analog-to-digitalconverter, in order to ensure that the sampling rate is equalto an integer multiple of the generation frequency given bythe standard. After the resampling operation, the beginningand the end of each subframe within the acquired signalare detected by searching for the gap that separates eachtwo successive subframes. The signal is next downsampledto the nominal generation frequency, and partitioned into aseries of OFDM symbols. After discarding the cyclic prefixwithin the OFDM symbols, a transformation into the frequencydomain is performed using Fast Fourier Transform (FFT). Notethat before we start joint I /Q and channel estimation, thetraining sequences in the first several OFDM symbols, i.e., the
386 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 3, MARCH 2017
Fig. 2. Flowchart of the proposed I /Q measurement methodology.
PREAMBLE field, are used to synchronize frequency/timingerrors [29].
By taking into consideration the channel and noise effects,the received discrete-time baseband signal at the measurementtestbed can be described in the frequency domain as
Yk(m) = Hk[βSk(m) + αS∗−k (m)] + Uk(m) (6)
where Hk is the frequency-selective channel impulse responseat the subcarrier k, and Uk(m) is assumed to be the zero-mean white Gaussian noise with equal variance σ 2
u for allsubcarriers.
B. Joint Coarse I /Q Imbalance Estimation and ChannelEqualization
To give a clear illustration of the proposed method,we choose 802.11ac 80-MHz very high throughput (VHT)waveforms [4] as an example; other OFDM-based WLANstandards, such as the IEEE 802.11a/g/n, are also applicable.The VHT long training field (VHT-LTF) occupies a total242 subcarriers out of 256, that is, {±2,±3, . . . ,±122}, while{−128 ∼ −123, 123 ∼ 127} and {±1, 0} are reserved asguard band and center subcarriers, respectively. The VHT shorttraining field (VHT-STF) is four times downsampled comparedwith VHT-LTF, occupying only 48 tones. In DATA field, thecollection of data subcarriers are almost the same as thoseof VHT-LTF subcarriers, except {±11,±39,±75,±103},which are used as pilot subcarriers. We shall denote thethree subcarrier collections of VHT-LTF, VHT-STF, and DATAfield as CV HT −LT F , CV HT −ST F , and CD AT A, respectively.
Using Yk(m) to represent received data in DATA field onsubcarrier k, from (6), we can now obtain the conjugate ofreceived data on mirror subcarrier −k as
Y ∗−k(m) = H ∗−k[β∗S∗−k(m) + α∗Sk(m)] + U∗−k(m). (7)
Define
λ1,k = αHk
β∗H ∗−k(8)
and by observing that the mirror frequency term in (6) canbe suppressed by subtracting Y ∗−k(m) multiplied by λ1, thisyields
Zk(m) = Yk(m) − λ1,kY ∗−k(m)
= w1,k Sk(m) + Uk(m) − λ1,kU∗−k(m) (9)
where
w1,k = |β|2 − |α|2β∗ Hk (10)
and Zk(m) is the data on subcarrier k with eliminated MFI.In a similar way, the conjugate of the signal on
subcarrier −k, with removed MFI, can be derived as
Z∗−k(m) = Y ∗−k(m) − λ2,kYk(m)
= w2,k S∗−k(m) + U∗−k(m) − λ2,kUk(m) (11)
where
λ2,k = α∗ H ∗−k
β Hk(12)
and
w2,k = |β|2 − |α|2β
H ∗−k. (13)
From the analysis in (9) and (11), in order to extract theestimate Sk(m) which is equalized and with removed I /Qimbalance, we need appropriate estimates for λ1,k , λ2,k , w1,k ,and w2,k . This will allow us to compute the exact amounts ofamplitude/phase imbalances from (3), since
γ = β
α
= w1,k
w2,kλ1,kor
(w2,k
w1,kλ2,k
)∗. (14)
A straightforward estimation on w1,k and w2,k can be achievedby using (9) and the VHT-LTF part of received signal ata VSA, as
w1,k = Zk,V HT −LT F
Sk,V HT −LT F= Yk,V HT −LT F − λ1,kY ∗
k,V HT −LT F
Sk,V HT −LT F
w2,k = w∗1,−k, k ∈ CD AT A. (15)
To obtain λ1,k or λ2,k , we here propose a novel blind methodwhich exploits the full second-order statistics of Yk(m) andits mirror frequency counterpart Y−k(m). For the receivedsignals in DATA field, the second-order moments of Yk(m)and Y−k(m), denoted as a and b, respectively, are defined as
a = E[Yk(m)Y ∗k (m)]=(|β|2+|β|2)|Hk|2σ 2
s + σ 2u (16)
b = E[Y−k(m)Y ∗−k(m)]=(|β|2+|β|2)|H−k|2σ 2s + σ 2
u . (17)
LI et al.: NONCIRCULAR MEASUREMENT AND MITIGATION OF I /Q IMBALANCE FOR OFDM-BASED WLAN TRANSMITTERS 387
Since high signal-to-noise ratio (SNR) conditions are expectedat the VSA side at the instrumentation level, to remove thenoise terms in both (16) and (17), we calculate
a ≈ (|β|2 + |β|2)|Hk|2σ 2s (18)
b ≈ (|β|2 + |β|2)|H−k|2σ 2s . (19)
In addition, due to the impropriety of the modulator outputs atthe transmitter, that is, Xk(m) and X−k(m), the pseudocrosscorrelation between the received signal Yk(m) and its mirrorfrequency counterpart Y−k(m) does exist, which provides uswith another degree of freedom to access the second-orderstatistical behavior, denoted by c and given by
c = E[Yk(m)Y−k(m)] = 2βαHk H−kσ2s . (20)
We can now define a new parameter
p = |c|2ab
(21)
which, according to (18)–(20), can be further evaluated as
p = 4|β|2|α|2|Hk|2|H−k|2σ 4s
(|β|2 + |α|2)2|Hk|2|H−k|2σ 4s
= 4|β|2|α|2(|β|2 + |α|2)2 . (22)
Upon substituting γ = βα into (22), and after a few mathemati-
cal manipulations, a fourth-order equation in |γ | is obtained as
p|γ |4 + (2 p − 4)|γ |2 + p = 0. (23)
Solving (23) for |γ |, and considering that |γ | ≥ 0, we obtaintwo solutions
|γ1| =√
(2 + 2√
1 − p)
p− 1, |γ2| =
√(2 − 2
√1 − p)
p− 1.
(24)
A feasible choice between the two solutions of |γ | is obtainedin the following.
Lemma 1: The range for |γ | is |γ | > 1 for all practicalvalues of |θ | < π/2.
Proof: Using the definitions of β and α in (2), we have|β|2 = 1 + g2 + 2gcos(θ) and |α|2 = 1 + g2 − 2gcos(θ). Thisproves the lemma, because cos(θ) > 0 for |θ | < π/2, andhence, |β| > |α| and |γ | > 1. �
Lemma 2: The ranges for |γ1| and |γ2| are |γ1| ≥ 1and |γ2| ≤ 1.
Proof: First, according to (23), |γ1|2 and |γ2|2 are reciprocalpairs, i.e., |γ2|2 = 1/|γ1|2, and so are |γ1| and |γ2|, meaningthat either |γ1| ≥ 1 and |γ2| ≤ 1, or |γ1| ≤ 1 and |γ2| ≥ 1.Then, from (21), observe that 0 < p ≤ 1, since
ab = (|β|2 + |β|2)2|Hk|2|H−k|2σ 4s
≥ 4|β|2|β|2|Hk|2|H−k|2σ 4s = |c|2. (25)
Consider now the derivative of |γ1|2 with respect to p,given by
d|γ1|2dp
= −1√1 − p
− 2 + 2√
1 − p
p2
= −(1 − p)2 − 2√
1 − p − 1
p2√
1 − p. (26)
For p ∈ (0, 1), (d|γ1|2)/(dp) < 0, |γ1|2 is a monotonousdecreasing function of p, for which the minimum value is 1only if p = 1. Therefore, |γ1| ≥ 1, and hence, |γ2| ≤ 1 �
From Lemmas 1 and 2, it can be concluded that the feasiblesolution for |γ | is |γ1|, is given by
|γ | =√
(2 + 2√
1 − p)
p− 1. (27)
Subsequently, λ1,k and λ2,k can be expressed as
λ1,k = 1
2
(1 + 1
|γ |2)
c
b(28)
λ2,k = 1
2
(1 + 1
|γ |2)( c
a
)∗. (29)
Now that |γ |, λ1,k , and λ2,k have been expressed asa function of the ensemble averages a, b, and c, theirestimates based on a single realization of the signals{Yk(m) | m = 0, 1, . . . , M − 1} and {Y−k(m) | m = 0, 1, . . . ,M − 1} can be derived by exploiting the ergodic propertyof Xk(m) and X−k(m). We are, therefore, able to obtainconsistent mean square estimators for a, b, and c in the form
aM = 1
M
M−1∑m=0
|Yk(m)|2
bM = 1
M
M−1∑m=0
|Y−k(m)|2
cM = 1
M
M−1∑m=0
Yk(m)Y−k(m) (30)
where M represents the total number of received OFDM sym-bols in DATA field. Then, as Xk(m) and X−k(m) are ergodic ina wide sense, so are Yk(m), Y−k(m), |Yk(m)|2, |Y−k(m)|2, andYk(m)Y−k(m); thus, limM→∞ aM = a, limM→∞ bM = b, andlimM→∞ cM = c. Therefore, λ1,k and λ2,k can be explicitlyestimated as
λ1,k =cM (1 +
√1 − |cM |2
aM bM)
bM (2 + 2
√1 − |cM |2
aM bM− |cM |2
aM bM)
(31)
λ2,k =c∗
M (1 +√
1 − |cM |2aM bM
)
a∗M (2 + 2
√1 − |cM |2
aM bM− |cM |2
aM bM)
(32)
and we have limM→∞ λ1,k = λ1,k and limM→∞ λ2,k =λ2,k [30]. Now, with w1,k , w2,k , λ1,k , and λ2,k in hand, coarseestimation of γ can be achieved using (14), and from (9),
388 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 3, MARCH 2017
the signal in DATA field, Sk(m), which is channel equalizedand with coarsely removed MFI, can be obtained as
Sk(m) = Yk(m) − λ1,kY ∗−k(m)
w1,k. (33)
C. Fine I /Q Imbalance Estimation
In order to enhance measurement accuracy based on theinitial estimates, we can now apply the standard hard-decision detection [16], [26] on Sk(m) to obtain the recoveredsignal Sk(m) according to the constellation mapping, whichwill be reused for fine I /Q imbalance estimation. Define thereceived signal vector in DATA field as
yk = [Yk(0), Yk(1), . . . , Yk(M)]T (34)
and the demodulated data matrix as
Sk =
⎡⎢⎢⎢⎣
Sk(0) S∗−k(0)
Sk(1) S∗−k(1)...
...
Sk(M − 1) S∗−k(M − 1)
⎤⎥⎥⎥⎦ (35)
to give
yk = Skgk (36)
where gk is the joint I /Q imbalance and channel impulseresponse parameter vector, defined as gk = [β Hk, αHk]T , forwhich the optimal estimator in the LSs sense is given by
gk = (SHk Sk)
−1SHk yk (37)
and a fine estimator of γ as
γ = 1
234
∑k∈CD AT A
gk(1)
gk(2)(38)
where 234 is the total number of available subcarriersin CD AT A , and the fine-estimated I /Q amplitude and phaseimbalances, that is, g and θ , can be obtained using thedefinition γ = (β)/(α) and (2), since
γ = β
α= 1 + ge− jθ
1 − ge− jθ. (39)
IV. SIMULATIONS AND EXPERIMENTAL RESULTS
In order to illustrate the performance of the proposed I /Qimbalance measurement method, simulations in the MATLABprogramming environment and real world experiments wereconducted. For comparison, a standard-compliant block-basedI /Q imbalance estimation technique from the literature wasalso considered [26]. This method, called the post-FFT LSs,was originally designed for I /Q imbalance measurement forOFDM receivers and has been merely customized for OFDMtransceivers by considering a different mathematical modelof I /Q imbalance to calculate the amplitude imbalance gand the phase imbalance θ using (2), as compared with theone used for OFDM receivers. This method has a similarimplementation as the proposed one. The difference lies in theproposed joint I /Q and channel estimation step, where LSs are
performed with the aid of a training sequence. However, sincethe standard dedicated training sequence VHT-STF occupiesonly a subgroup of the subcarriers, in order to equalize thereceived OFDM symbols in the DATA field, interpolation andextrapolation on the estimated channel impulse response arerequired within the fine I /Q imbalance estimation step. Formore detail on its implementation, we refer to [26].
A. Computer Simulations
We first simulated an OFDM-based WLAN transmissionsystem fully compliant with the IEEE 802.11 ac. The discretebaseband waveform s(n) was generated with the followingsystem parameters: length of subcarriers K = 256, length ofcyclic prefix Kcp = 64, waveform bandwidth Bc = 80 MHz,and therefore, the OFDM symbol duration Tsym =(K + Kcp)/Bc = 4 μs, and guard interval Tg = Kcp/Bc =0.8 μs. The amplitude and phase mismatches imposed on s(n)were set to g = 1.3 and θ = 7◦. The transmission channel con-sidered was a three-tap multipath static-Rayleigh channel withfrequency selectivity, where h(n) = [0.866 + 0.5j, 0.0643 +0.0766j, 0.0098−0.0017j ] (symbol period spacing) [31], anddifferent levels of the white Gaussian noise u(n) were added.
In the coarse estimation stage, the proposed methodemployed M = 3000 OFDM symbols in DATA field tocalculate the full second-order statistics of the receiveddiscrete-time baseband signal Yk(m) and its mirror frequencycomponent Y−k(m), that is, aM , bM , and cM , as given in (30).These were employed for the estimation of λ1,k from (31)along each subcarrier k ∈ CD AT A. By using the standard-compliant VHT-LTF part of received signal and λ1,k in hand,we estimated the parameter w1,k according to (15), and sub-sequently the coarsely MFI-eliminated and channel-equalized
signal Sk(m) from (33) for each subcarrier k ∈ CD AT A.In the fine estimation step, first 200 symbols of Sk(m) wererecalibrated by a hard-decision detector to obtain the recoveredsignal Sk(m). Next, from the recovered signal Sk(m) and thereceived signal Yk(m), a fine measurement of γ was performedin the LS sense by using (37) and (38), and from (39),we obtained fine estimate of both the amplitude and phaseimbalances.
The performance evaluation was performed through theestimation bias in both the amplitude and phase imbalances,quantified as
g = 20 log10
(g − g
g
)and θ = 20 log10
(θ − θ
θ
)
(40)
where g and θ are, respectively, the estimates of g and θ ,obtained by averaging 1000 independent trails.
Two sets of simulations were carried out based on differentconstellation sizes. In the first one, a 64-QAM modulationscheme conforming with the MCS 5 of the IEEE 802.11acstandard was employed. As shown in Fig. 3(a) and (b), theestimation performances of the post-FFT LS method andthe proposed second-order statistics-based one were simi-lar in the high SNR region for the 64-QAM modulationscheme (MSC5), but the proposed method was able to more
LI et al.: NONCIRCULAR MEASUREMENT AND MITIGATION OF I /Q IMBALANCE FOR OFDM-BASED WLAN TRANSMITTERS 389
Fig. 3. Estimation performances of the post-FFT LS method [26] and the proposed method for a multipath (frequency selective) channel based on the IEEE802.11ac 80-MHz 64-QAM modulation (MCS 5). (a) Amplitude bias g. (b) Phase bias θ .
Fig. 4. Estimation performances of the post-FFT LS method [26] and the proposed method for a multipath (frequency selective) channel based on the IEEE802.11ac 80-MHz 256-QAM modulation (MCS 8). (a) Amplitude bias g. (b) Phase bias θ .
accurately estimate the I /Q impairment in both the amplitudeand phase in the lower SNR region. This was mainly due to thefact that the required channel interpolation and extrapolationoperations within the post-FFT LS method were sensitive tolarge noise levels. The proposed method can achieve directchannel equalization on each subcarrier in DATA field, whichfacilitates a more accurate hard-decision operation and wasmore accurate to extract recovered data for the subsequent fineI /Q imbalance estimation. As expected, these performanceadvantages were more pronounced when higher order MCSswere employed, e.g., a 256-QAM modulation (MCS 8), asshown in Fig. 4. However, we should mention that thisenhanced reliability is achieved at a cost of increased com-putational complexity, because more observations are requiredto make the moment statistics in (30) mean square consistent.By considering Figs. 3 and 4, it is also interesting to observethat both methods gave better estimation of the amplitudeimbalance g than the phase imbalance θ at the same SNR.This can be explained by Fig. 5, where as compared with g,
θ was the major cause of estimation bias on γ , which is thekey parameter to calculate g and θ and was obtained by theproposed method in its fine I /Q imbalance estimation stage.This was also the case with the post-FFT LS method, and itmay result from the nonlinear relationship of γ on g and θ ,as given in (39).
B. Experimental Results
An experimental setup was developed to further evaluatethe performance of the proposed second-order statistics-basedI /Q imbalance measurement method. The measurementtestbed built upon Aeroflex PXI modules is shown in Fig. 6,and the description of each PXI module is given in Table I.In the setup, the baseband I /Q waveforms were created byAeroflex IQ Creator software running on a PC. The modulatoroutput signals were then loaded into a nonvolatile memory ofthe vector signal generator (VSG). After low-pass filtering, theI /Q signal waveforms were fed to a quadrature modulator to
390 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 3, MARCH 2017
Fig. 5. Sensitivities of g and θ versus γ obtained by the proposed method at SNRs ranging from 15 to 45 dB. (a) 64-QAM modulation (MCS 5).(b) 256-QAM modulation (MCS 8).
Fig. 6. I /Q imbalance measurement setup based on Aeroflex PXI modules.
generate a carrier-modulated RF signal at chan-nel 36 (5.18 GHz). A VSA was used to detect the transmittedRF signal in a loopback way [32] and to down-convertedit to baseband. After I /Q demodulation, complex-valuedbaseband data were transferred to the same PC, where I /Qimbalance estimation was implemented. Both the VSG andVSA were synchronized by the same 10-MHz referenceclock. The RF output level of VSG was set to −10 dBm, andthe reference level of VSA was set to 0 dBm. The OFDMsystem parameters, such as waveform bandwidth, OFDMsymbol duration, and guard interval, were the same as thosein the previous computer simulations. The sampling rate atthe VSA side was fs = 160 MHz.
Three types of signals, fully compliant with the IEEE802.11ac 80 MHz, were generated. The first one employeda low density QPSK modulation (MCS 1) without any I /Q
TABLE I
DESCRIPTIONS OF AEROFLEX PXI MODULES
TABLE II
ESTIMATED VARIANCES OF THE EQUIVALENT BASEBAND CHANNEL
TRANSFER FUNCTION IN BOTH TIME AND FREQUENCY DOMAINS
imbalance, and was used to estimate the equivalent basebandchannel effect at this instrumentation level measurement. Thelength of OFDM symbols in DATA field was set to 3000.These OFDM symbols were known to the VSA, and hence,a direct measurement of the channel impulse response couldbe performed, resulting in a 3000 × 234 channel transferfunction matrix Hn,m. We calculated the variances of thechannel transfer function in both time (per symbol period)and frequency (per subcarrier) along the two dimensions ofthis matrix; the results are given in Table II. Observe thatcompared with the time variation, the frequency-selectivityin channel, which is further shown in Fig. 7, is the domi-nant effect. This observation justifies our frequency-selectiveassumption on the channel, and illustrates the suitability of
LI et al.: NONCIRCULAR MEASUREMENT AND MITIGATION OF I /Q IMBALANCE FOR OFDM-BASED WLAN TRANSMITTERS 391
Fig. 7. Normalized channel impulse response per subcarrier spacing; the results were obtained by averaging 3000 OFDM symbols. (a) Normalized amplitude–frequency response of channel averaged in time. (b) Normalized phase–frequency response of channel averaged in time.
TABLE III
I /Q MEASUREMENT RESULTS
the proposed joint I /Q imbalance measurement and channelequalization scheme. It also indicates a potential way toimprove the accuracy of I /Q imbalance measurement methodsfor transmitters [14]–[16], where an ideal baseband mea-surement channel is assumed. Note that although the channelimpulse response is also assumed to be time-invariant, a slightfluctuation is inevitable in real world measurements.
In the next experiment, MCS 5 and MCS 8 modulatedwaveforms with different amounts of amplitude and phaseimbalances were loaded into VSG. The measurement resultsare given in Table III. Again, observe that in all the cases theamplitude imbalance was more accurately estimated by bothconsidered methods as compared with the phase imbalance.According to its specifications [33], for this experiment, theinstrument was very likely to work in a high SNR environment.The performance advantages of the proposed second-orderstatistics-based method over the post-FFT LS method were
not obvious for the 64-QAM modulation (MSC 5); however,the proposed method was able to more accurately estimate theimpairment caused by the I /Q imbalance when higher orderMCSs were employed, e.g., a 256-QAM modulation (MSC 8),with a 1–4 dB and 2–8 dB performance improvement inrespective amplitude and phase imbalance estimation.
V. CONCLUSION
A novel two-stage I /Q imbalance measurement methodfor OFDM-based WLAN transmitters at the instrumentationlevel has been proposed. In the initial stage, the noncircu-lar second-order statistics, assisted by the standard-complianttraining sequences, has enabled the decoupling of the I /Qimbalance distortion from the effects of the measurementframework. This has made it possible to account for frequencyselectivity of the channel at the instrumentation level for
392 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 3, MARCH 2017
wideband multicarrier systems. In the next stage, an enhancedI /Q imbalance measurement has been performed using astandard decision-directed scheme, in order to recalibratethe initially recovered symbols transmitted in DATA field ofstandardized OFDM systems. It has been further demonstratedthrough computer simulations and real world experiments onsignals fully compliant with the IEEE 802.11 ac 80 MHzthat the proposed method exhibits enhanced measurementrobustness against both additive noise and frequency-selective transmission, as compared with the state-of-art post-FFT LS method. This advantage is more pronounced forhigher order modulation schemes, such as the 256-QAMmodulation (MSC 8).
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Zhe Li (S’16) received the B.S. degree in telecom-munication engineering from the Nanjing Universityof Posts and Telecommunication, Nanjing, China, in2011, and the M.S. degree in software engineeringfrom Southeast University, Nanjing, in 2014, wherehe is currently pursuing the Ph.D. degree with theSchool of Information and Engineering. His currentresearch interests include complex-valued statisticalanalysis and its applications on I/Q measurement.
LI et al.: NONCIRCULAR MEASUREMENT AND MITIGATION OF I /Q IMBALANCE FOR OFDM-BASED WLAN TRANSMITTERS 393
Yili Xia (M’11) received the B.Eng. degree ininformation engineering from Southeast University,Nanjing, China, in 2006, and the M.Sc. degree(Hons.) in communications and signal processingfrom the Department of Electrical and ElectronicEngineering, Imperial College London, London,U.K., in 2007, and the Ph.D. degree in adaptivesignal processing from the Imperial College Londonin 2011.
Since 2013, he has been an Associate Professorwith the School of Information and Engineering,
Southeast University. His current research interests include complex-valuedlinear and nonlinear adaptive filters and complex-valued statistical analysisand their applications on communications.
Wenjiang Pei received the M.S. and Ph.D. degreesin instrumentation and measurement from theNanjing University of Aeronautics and Astronautics,Nanjing, China, in 1995 and 1997, respectively.
He is currently a Professor with the School ofInformation and Engineering, Southeast University,Nanjing. His research interests signal processing andhardware instrumentation for communications.
Kai Wang received the Ph.D. degree in signalprocessing from the School of Information and Engi-neering, Southeast University, Nanjing, China, in2009.
He is currently an Associate Professor in signalprocessing with Southeast University. His currentresearch interests include parameter estimation andsignal processing for communications.
Yongming Huang (M’10) received the B.S. andM.S. degrees from Nanjing University, Nanjing,China, in 2000 and 2003, respectively, and the Ph.D.degree in electrical engineering from Southeast Uni-versity, Nanjing, in 2007.
Since 2007, he has been a Faculty Member withthe School of Information Science and Engineer-ing, Southeast University, where he is currently aFull Professor. From 2008 to 2009, he visited theSignal Processing Laboratory, School of ElectricalEngineering, KTH Royal Institute of Technology,
Stockholm, Sweden. His current research interests include multiple-antennawireless communications and signal processing.
Dr. Huang serves as an Associate Editor of the IEEE TRANSACTIONSON SIGNAL PROCESSING, the EURASIP Journal on Advances in SignalProcessing, and the EURASIP Journal on Wireless Communications andNetworking.
Danilo P. Mandic (M’99–SM’03–F’12) received thePh.D. degree in nonlinear adaptive signal processingfrom the Imperial College London, London, U.K.,in 1999.
He is currently a Professor in signal process-ing with the Imperial College London. He hasbeen a Guest Professor with Katholieke Univer-siteit Leuven, Leuven, Belgium, the Tokyo Univer-sity of Agriculture and Technology, Tokyo, Japan,and Westminster University, London, and a Fron-tier Researcher with RIKEN, Wako, Japan. He has
authored the book Recurrent Neural Networks for Prediction: LearningAlgorithms, Architectures and Stability (First Edition, 2001) and ComplexValued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and NeuralModels (First Edition, Wiley, 2009). He was an Editor of the book SignalProcessing Techniques for Knowledge Extraction and Information Fusion(Springer, 2008) and over 200 publications on signal and image processing.His current research interest include in the area of nonlinear adaptive signalprocessing, multivariate data analysis, and nonlinear dynamics.
Dr. Mandic has been a member of the IEEE Technical Committee on SignalProcessing Theory and Methods. He has been an Associate Editor of the IEEESignal Processing Magazine, the IEEE TRANSACTIONS ON CIRCUITS AND
SYSTEMS II, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, the IEEETRANSACTIONS ON NEURAL NETWORKS, and the International Journal ofMathematical Modeling and Algorithms. He has produced award winningpapers and products resulting from his collaboration with industry.
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